A sample space is a list or grid of all the possible outcomes of an experiment. For two events, such as rolling two dice, a grid shows every combination: 6 × 6 = 36 equally likely outcomes. Count the outcomes that satisfy the event and divide by the total to find the probability.
Sample spaces
- Total outcomes = outcomes of first event × outcomes of second event
- Two dice: 36 outcomes; a coin and a die: 12; three coins: 8
- P(event) = favourable outcomes ÷ total outcomes
Listing outcomes
List all the possible outcomes systematically, so none are missed. For three coins, list HHH, HHT, HTH, HTT, THH, THT, TTH and TTT, which is 8 outcomes. For a coin and a die, pair each side of the coin with each number: H1 to H6, then T1 to T6, giving 12 outcomes.
Two-way grids
For two dice, draw a 6 by 6 grid with one die across the top and the other down the side, and fill each cell with the result you need, such as the total, the difference or the product. Each of the 36 cells is equally likely, so counting cells gives probabilities directly.
Finding probabilities
Count the cells or outcomes that match the event and divide by the total. For two dice, a total of 9 occurs 4 times (3 + 6, 4 + 5, 5 + 4, 6 + 3), so P(total of 9) = 4/36 = 1/9.
Different combining rules
The grid can show any combination, not just totals. For the difference between two dice, a difference of 2 appears 8 times out of 36, giving 8/36 = 2/9. For two spinners numbered 1 to 4 and 1 to 5, the grid has 20 cells.
Combined events
Questions about “and” or “or” can be answered by counting in the sample space. For a coin and a die, P(tail or a 6) counts all six tails plus the head with a 6: 7 outcomes out of 12, so 7/12. Be careful not to count the tail with a 6 twice.
Worked examples
Example 1. Three fair coins are flipped. Find P(exactly two heads).
- List the 8 outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
- Exactly two heads: HHT, HTH, THH.
Answer: 3/8.
Example 2. Two fair dice are rolled and the scores added. Find P(total of 9).
- Outcomes giving 9: (3, 6), (4, 5), (5, 4), (6, 3).
- 4 out of 36.
Answer: 4/36 = 1/9.
Example 3. Two fair dice are rolled and the difference between the scores is found. Find P(difference of 2).
- Pairs with difference 2: (1,3), (2,4), (3,5), (4,6) and the reverse, 8 in total.
Answer: 8/36 = 2/9.
Example 4. A fair coin is flipped and a fair die rolled. Find P(tail or 6).
- Sample space: 12 outcomes.
- All six tails, plus H6: 7 outcomes.
Answer: 7/12.
Example 5. Two spinners, one numbered 1 to 4 and one numbered 1 to 5, are spun and the numbers added. Find P(total of 6).
- Total outcomes: 4 × 5 = 20.
- Totals of 6: (1, 5), (2, 4), (3, 3), (4, 2).
Answer: 4/20 = 1/5.
Common mistakes
- Missing outcomes when listing. Use a fixed order or a grid.
- Counting (3, 6) and (6, 3) as the same. They are different outcomes, one for each die.
- Using 12 outcomes for two dice. Two dice have 6 × 6 = 36 outcomes.
- Double-counting in “or” questions. Count each matching outcome only once.
- Forgetting to simplify. Simplify the fraction if the question asks.
How sample space diagrams are examined
Sample space diagrams are Foundation probability content for both 1MA1 and 4MA1, and they are often the first part of a longer probability question. Questions ask you to:
- list all the outcomes of two events systematically
- complete a two-way grid for two dice or spinners
- find probabilities from a sample space
- answer “and” and “or” questions by counting outcomes
The sample space worksheet has questions with answers, and the GCSE probability revision engine gives more.
Listing outcomes uses systematic listing. For events in stages, see tree diagrams.
Frequently asked questions
What is a sample space?
A list or grid of all the possible outcomes of an experiment.
How many outcomes are there for two dice?
36, because each of the 6 outcomes of one die can pair with each of the 6 of the other.
How do you find a probability from a sample space?
Count the outcomes that match the event and divide by the total number of outcomes.